Let \(J(t) = \int _{\mathbb {S}^{n-1}} \phi (h(x,t))f(x)\mathop {}\!\mathrm {d}x\) . We claim that \(J(t)\)  is unchanged along the flow \((\ref {feq})\) . It is because
\(\begin{split}J^{\prime }(t) &= \int _{\mathbb {S}^{n-1}}\phi ^{\prime }(h)\partial _{t} hf(x) \mathop {}\!\mathrm {d}x \\&= \int _{\mathbb {S}^{n-1}}\frac{f(x)}{\varphi (h)}\partial _{t} h \mathop {}\!\mathrm {d}x \\&= \int _{\mathbb {S}^{n-1}}\frac{f(x)}{\varphi (h)}\left(\frac{1}{f(x)} \sigma _{k}(x)\varphi (h)h\eta (t) - h\right)\mathop {}\!\mathrm {d}x\\&= \int _{\mathbb {S}^{n-1}}\sigma _{k}(x)h\eta (t)\mathop {}\!\mathrm {d}x - \int _{\mathbb {S}^{n-1}}\frac{h}{\varphi (h)}f(x) \mathop {}\!\mathrm {d}x\\& = 0.\end{split}\) 
For each \(t \in [0,T)\) , suppose that the maximum of radial function \(\rho (\cdot ,t)\)  is attained at some \(u_{t} \in \mathbb {S}^{n-1}\) .
Let
\(R_{t} = \max _{u \in \mathbb {S}^{n-1}}\rho (u,t) = \rho (u_{t},t)\) 
for some \(u_{t}\in \mathbb {S}^{n-1}\) .
By the definition of support function, we have
\(h(x,t)\ge R_t\langle x,u_t \rangle , \quad \forall x\in \mathbb {S}^{n-1}.\) 
Denote the hemisphere containing \(u_t\)  by \(S_{u_{t}}^{+} =\left\lbrace x\in \mathbb {S}^{n-1}: \langle x,u_t \rangle > 0\right\rbrace \) . Since \(\phi ^{\prime }(h) = \frac{1}{\varphi (h)} > 0 \)  implies that \(\phi (h) \)  is strictly increasing about \(h\) , we have
\(\begin{split}J(0) & = J(t)\\& \ge \int _{S_{u_{t}}^{+}} \phi (h(x,t))f(x)\mathop {}\!\mathrm {d}x \\& \ge \int _{S_{u_{t}}^{+}}\phi (R_{t} \langle x,u_t \rangle )f(x)\mathop {}\!\mathrm {d}x\\& \ge f_{\min }\int _{S_{u_{t}}^{+}}\phi (R_{t} \langle x,u_t \rangle )\mathop {}\!\mathrm {d}x\\&= f_{\min } \int _{S^{+}} \phi (R_tx_1) \mathop {}\!\mathrm {d}x,\end{split}\) 
where \(S^{+} =\left\lbrace x\in \mathbb {S}^{n-1}: x_1\ > 0\right\rbrace \) .
Denote \(S_1 =\left\lbrace x\in \mathbb {S}^{n-1}: x_1\ge 1/2\right\rbrace \) , then
\(\begin{split}J(0)&\ge f_{\min } \int _{S_1} \phi (R_t/2) \mathop {}\!\mathrm {d}x \\&= f_{\min } \phi (R_t/2) |S_1|,\end{split}\) 
which implies that \(\phi (R_t/2)\)  is uniformly bounded from above.
By assumption (A), \(\phi (s)\)  is strictly increasing and tends to \(+\infty \) 
as \(s\rightarrow +\infty \) .
Thus \(R_t\)  is uniformly bounded from above.
Now we can prove that \(\eta (t)\)  has positive lower bound. Since mixed volumes are monotonic increasing, see [1]}, we
have for each \(t \in [0,T)\)
\(h_{\min }^{k+1}(t) \le \dfrac{ \int _{\mathbb {S}^{n-1}}h\sigma _{k} \mathop {}\!\mathrm {d}x }{\omega _{n-1}}\le h_{\max }^{k+1}(t),\) 
here \(h_{\min }(t) = \min _{x \in \mathbb {S}^{n-1}} h(x,t)\)  and \(h_{\max }(t) = \max _{x \in \mathbb {S}^{n-1}} h(x,t)\) .
This together with Lemma REF  and the upper bound of \(h\)  implies that there exist positive constants \(c_{1}\)  and \(c_{2}\)  such that
\(\int _{\mathbb {S}^{n-1}}h\sigma _{k} \mathop {}\!\mathrm {d}x \le c_{1}\) 
and
\(h_{\max }(t) \ge c_{2}.\) 
Recalling the definition of \(\eta (t) \)  and noticing that \(\dfrac{1}{\varphi (s)}\)  is an increasing function,
we have
\(\begin{split}\eta (t) & = \dfrac{\int _{\mathbb {S}^{n-1}}hf(x)/\varphi (h)\mathop {}\!\mathrm {d}x}{\int _{\mathbb {S}^{n-1}}h\sigma _{k}\mathop {}\!\mathrm {d}x}\\&\ge \frac{1}{c_{1}}\int _{S_{u_{t}}^{+}} hf(x)/\varphi (h)\mathop {}\!\mathrm {d}x\\& \ge \frac{1}{c_{1}}\int _{S_{u_{t}}^{+}}R_{t} \langle x,u_t \rangle f_{\min }\frac{1}{\varphi (R_{t}\langle x,u_t \rangle )}\mathop {}\!\mathrm {d}x\\& = \frac{1}{c_{1}}f_{\min }\int _{S^{+}}R_{t}x_{1}\frac{1}{\varphi (R_{t} x_{1})}\mathop {}\!\mathrm {d}x\\& \ge \frac{1}{c_{1}}f_{\min }|S_{1}| \frac{1}{2}R_{t}\frac{1}{\varphi ( \frac{1}{2}R_{t})}\\& \ge c_{3},\end{split}\) 
where \(c_{3}\)  is a positive constant independent of \(t\) .
Suppose the minimum of \(h(x,t)\)  is attained at a point \((x_{t}, t)\) . At \((x_{t}, t)\) , \(\nabla _{ij}h\)  is non-negative. It follows that
\(\sigma _{k}(x_{t}, t) \ge h_{\min }^{k}(t).\) 
Then in the sense of the lim inf of difference quotient, see [2]}, we have
\(\begin{split}\frac{\partial h_{\min }(t)}{\partial t} & \ge \frac{1}{f_{\max }}h_{\min }(t)[\eta (t)h_{\min }^{k}(t)\varphi (h_{\min }(t)) - f(x)]\\& \ge \frac{1}{f_{\max }}h_{\min }(t)[c_{3}h_{\min }^{k}(t)\varphi (h_{\min }(t))- f_{\max }].\end{split}\) 
If \(\varphi (s) > \alpha s^{-k-\varepsilon } \)  for some positive constant \(\varepsilon \)  for \(s \)  near 0, then
\(\begin{split}\frac{\partial h_{\min }(t)}{\partial t}& \ge \frac{1}{f_{\max }}h_{\min }(t)(h_{\min }^{-\varepsilon }(t)\alpha c_{3}- f_{\max }).\end{split}\) 
The right hand of the above inequality is positive for \(h_{\min }(t)\)  small enough and the lower bound of \(h_{\min }(t)\)  follows from the maximum principle in [2]}.
